Seeing through objects in Game engine
Your problem is very simple to fix. your normals are reversed.

You can easily see which faces are pointing inward by enabling Backface Culling. This enables you to see though the back of faces like you do when you press P.



You could also enter edit mode on that object, and enable the normals visualization.



On high-polly models such as this one, this may or may not help much.

to fix this you could manually go though and select each face that is pointing the wrong way, and flip the normals by pressing the  Flip Normals  button.



You Could also select all of your faces and press the  Recalculate  button, to recalculate the normals. This sometimes works, and sometimes it makes it worse. It is a good idea to try this first, because there is always Ctrl+Z for the times when it does not work.

In this case, it made all of the normals point the same way, but then they were all pointing the wrong direction. Simply flipping all of them could fix this easily enough. Sometimes more complex objects can confuse blender and recalculate them quite wrong.



For your sky object, see  this question